Henson et al, 2011 - Wellcome Trust Centre for Neuroimaging

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Review Article
published: 24 August 2011
doi: 10.3389/fnhum.2011.00076
HUMAN NEUROSCIENCE
A parametric empirical Bayesian framework for the EEG/MEG
inverse problem: generative models for multi-subject and
multi-modal integration
Richard N. Henson1*, Daniel G. Wakeman1, Vladimir Litvak2 and Karl J. Friston2
Cognition and Brain Sciences Unit, Medical Research Council, Cambridge, UK
Wellcome Trust Centre for Neuroimaging, University College London, London, England
1
2
Edited by:
Luis M. Martinez, Universidade da
Coruña, Spain
Reviewed by:
Srikantan S. Nagarajan, University of
California, USA
Nelson Jesús Trujillo-Barreto, Cuban
Neuroscience Centre, Cuba
Masa-aki Sato, ATR Neural Information
Analysis Laboratories, Japan
*Correspondence:
Richard N. Henson, Cognition and
Brain Sciences Unit, Medical Research
Council, 15 Chaucer Road, Cambridge
CB2 2EF, UK.
e-mail: [email protected]
We review recent methodological developments within a parametric empirical Bayesian (PEB)
framework for reconstructing intracranial sources of extracranial electroencephalographic
(EEG) and magnetoencephalographic (MEG) data under linear Gaussian assumptions. The
PEB framework offers a natural way to integrate multiple constraints (spatial priors) on this
inverse problem, such as those derived from different modalities (e.g., from functional magnetic
resonance imaging, fMRI) or from multiple replications (e.g., subjects). Using variations of the
same basic generative model, we illustrate the application of PEB to three cases: (1) symmetric
integration (fusion) of MEG and EEG; (2) asymmetric integration of MEG or EEG with fMRI,
and (3) group-optimization of spatial priors across subjects. We evaluate these applications on
multi-modal data acquired from 18 subjects, focusing on energy induced by face perception
within a time–frequency window of 100–220 ms, 8–18 Hz. We show the benefits of multi-modal,
multi-subject integration in terms of the model evidence and the reproducibility (over subjects)
of cortical responses to faces.
Keywords: source reconstruction, bioelectromagnetic signals, data fusion, neuroimaging
Introduction
Distributed approaches to the EEG/MEG inverse problem typically entail the estimation of ∼104 parameters, which reflect the
amplitude of dipolar current sources (in one to three orthogonal
directions) at discrete points within the brain, using data from
only ∼102 sensors that are positioned on (EEG), or a short distance from (MEG), the scalp (Mosher et al., 2003). For such illposed inverse problems, a Bayesian formulation offers a natural
way to introduce multiple constraints, or priors, to “regularize”
their solution (Baillet and Garnero, 1997). In particular, there is
growing interest in Bayesian approaches to integrate EEG and
MEG data, and data from functional magnetic resonance imaging (fMRI), in so-called multi-modal integration or “fusion”
schemes (e.g., Trujillo-Barreto et al., 2001; Sato et al., 2004;
Daunizeau et al., 2007; Ou et al., 2010; Luessi et al., 2011). Here,
we review our recent work in this area, in terms of a parametric
empirical Bayesian (PEB) framework that allows the fusion of
multiple modalities, and subjects, within the same generative
model.
The linearity of the electromagnetic forward model for E/
MEG (which maps each source to each sensor, based on quasistatic, numerical approximations to Maxwell’s equations)
means that the inverse problem can be expressed as a two-level,
hierarchical, probabilistic generative model, in which (hyper)
parameters of the higher level (sources) represent priors on the
parameters of the lower level (sensors). This hierarchical formulation allows the application of an Empirical Bayesian approach,
in which the hyperparameters that control the prior distributions can be estimated from the data themselves (see below, and
Frontiers in Human Neuroscience
Sato et al., 2004; Daunizeau et al., 2007; Trujillo-Barreto et al.,
2008; Wipf and Nagarajan, 2009; for related approaches). The
further assumption of Gaussian distributions for the priors and
parameters (hence “PEB,” Friston, et al., 2002) enables a simple
matrix formulation of the generative model in terms of covariance components (cf, Gaussian process modeling): Consider
the linear model
Y = LJ + E(1)
J = 0 + E(2) (1)
where Y ∈ n × t is a n (sensors) × t (time points) matrix of sensor
data; L ∈ n × p is the n × p (sources) “leadfield” or “gain” matrix,
J ∈ p × t is the matrix of unknown primary current density parameters, and E(l)∼N(0,C(l)) are random terms sampled from zero-mean,
multivariate Gaussian distributions1. For multiple trials, the data
can be further concatenated along the temporal dimension, for
example to accommodate induced (non-phase-locked) effects that
would be removed by trial-averaging (Friston et al., 2006).
A Generative Model
The spatial covariance matrices C(1) and C(2) can be represented by
a linear mixture of covariance components, Q(hl ):
( )
( )
C(l ) = cov E(l ) = ∑ h exp  h(l ) Qh(l ) (2)
1
In fact, the error covariance of these distributions is assumed to factorize into
temporal, V, and spatial, C, components (Friston et al., 2006), such that: vec(E(l ))
~N(0,V ⊗ C(l)), where vec vectorizes a matrix and ⊗ represents the Kronecker
tensor product. For simplicity, we will assume the temporal component (e.g., autocorrelations) are known, and focus here on optimizing the spatial components.
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August 2011 | Volume 5 | Article 76 | 1
Henson et al.
Parametric empirical Bayesian models for MEG/EEG
Basic generative (PEB) model
,
Q1(2) Q(2)
2
...
Q1(1) Q(1)
2
((1))
h
((2))
h
C((2))
...
C((1))
J
L
p() =  (, ) (4)
Y
Fixed
Variable
Data
Source and sensor space
Figure 1 | A Bayesian dependency graph representing the basic
generative model used for the EEG/MEG inverse problem for one subject
and one sensor-type (modality) within the Parametric Empirical Bayesian
(PEB) framework. The variables are defined in Eqs 2–4 in the text.
where  (hl ) is the “hyperparameter” for the h-th component of the
l-th level (and its exponentiation ensures positive covariance components). This results in the generative model illustrated by the
Bayesian dependency graph in Figure 1, which has the associated
probability densities:
p(Y | J,  (1) ) =  (LJ, C(1) ) p(J |  (2) ) =  (0, C(2) )
(3)
Note that the assumption of a zero-mean for the sources
in the second term in Eq. 3 means that C(2) functions as a
“shrinkage prior” on the sources. Indeed, it can be shown that
when the source covariance matrix is the identity matrix (i.e.,
Q(2) = Ip⇒C(2) = exp((2))Ip, where Ip is a p × p identity matrix),
the posterior (conditional) mean of J (see below) corresponds to
the standard “L2 minimum norm” (MMN), or Tikhonov, solution (Mosher et al., 2003; Hauk, 2004; Phillips et al., 2005). The
MMN solution explains data with minimal source energy, where
the relative weighting of data fit and source energy is controlled
by a regularization parameter, whose value is often derived from
an empirical estimate of the signal-to-noise ratio (Fuchs et al.,
1998). An elegant aspect of the above Empirical Bayesian formulation is that the degree of regularization is controlled by the
hyperparameters,  (hl ), whose values are optimized in a principled
manner based on maximizing (a bound on) the model evidence
(see below, and Phillips et al., 2005).
Furthermore, the general formulation above allows multiple
spatial priors, on both the sources and the sensors. At the sensor-level, for example, one could assume separate white-noise
(1)
components (Q1 = In ) for each type of sensor (see Application 1
below), and/or further empirical estimates of (non-brain) noise
sources; e.g., from empty-room MEG recordings (Henson et al.,
2009a). At the source level, one could add additional smoothness priors (Sato et al., 2004; Friston et al., 2008), or in a more
extreme case, one could use hundreds of cortical “patches”
within the solution space, to encourage a sparse solution. The
latter “multiple sparse priors” (MSP) approach (Friston et al.,
2008) entails Qh(2) = q h q hT , where qh ∈ ¡p × 1 are sampled from
a p × p matrix that encodes the proximity of sources within
the cortical mesh.
With many hyperparameters (and a potentially complex cost
function for gradient ascent; see below), it becomes prudent to
further constrain their values, using normal hyperpriors:
where  is a vector of hyperparameters concatenated over source
and sensor levels. A (weak) shrinkage hyperprior can be implemented by setting  = −4 (i.e., a small prior mean of exp(−4)
for each covariance component; though strictly speaking, the logcovariance has prior mean of −4) and  = 16 × I (i.e., a large prior
variance, allowing exp( (hl ) ) to vary over several order of magnitude;
Henson et al., 2007). This furnishes a sparse distribution of hyperparameters, where those that are not helpful (in maximizing the
model evidence) shrink to zero; such that their associated covariance components Q(hl ) are effectively switched off. This is a form of
automatic model selection (Friston et al., 2007), in that the optimal
model will comprise only those covariance components with nonnegligible hyperparameters2.
Model inversion
The optimal (conditional) mean and covariance of the hyperpara
ˆ are found using a Variation Bayesian approach under
meters (ˆ , )
the Laplace approximation (Friston et al., 2007). This maximizes a
cost function called the variational “free-energy,” under the Laplace
assumption:
ˆ = max arg  (, , )
ˆ , ,
The free-energy is related to the log of the model evidence; i.e.,
given a generative model M defined by Eqs 2–4, then:
∧ ∧
ln p ( | M ) = ln ∫∫ p ( , J, | M ) d Jd ≈   , , Y


If the generative model were linear in the (hyper) parameters,
this approximate equality would be exact. However, in our case,
the model is effectively a Gaussian process model and is therefore
non-linear in the hyperparameters. This renders the free-energy a
lower bound on log-evidence, but one that has been shown to be
effective in selecting between models of the present type (Friston
et al., 2007). For the generative model in Figure 1, the free-energy
is (ignoring constants):
2
Note that, although the prior mean and variance of these hyperpriors encourage a
sparse distribution of hyperparameters, such hyperpriors do not necessarily furnish
a sparse distribution of parameters, i.e., a sparse distribution of source estimates.
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Henson et al.
(
)
 (, , Y ) = −tr C −1YYT − ln | C | − ( - ) −1 ( - ) + ln −1
T
horizontal and vertical visual angles of approximately 3.66° and
5.38°. As soon as they saw the face or scrambled face, subjects used
(5) either their left or right index finger to report whether they thought
where C = LC(2)LT + C(1), C(l) = Sh exp αˆ (hl ) Q(hl ), and ,Ω are the it was symmetrical or asymmetrical about a vertical axis through its
means and variances respectively of the (hyper)prior distributions center (having been instructed that this judgment corresponded to
of hyperparameters, as in Eq. 4 (see Friston, et al., 2007, for details). more or less symmetrical “than average,” where an idea of average
The free-energy can also be considered as the difference between the symmetry was obtained from exposure to practice stimuli). This
model accuracy (a, the first two terms) and the model complexity task was chosen because it can be performed equally well for faces
(c, the second two terms). Heuristically, the first two terms rep- and scrambled faces.
resent the probability of the data under the assumption they have
There were 300 different faces and 150 different scrambled
a covariance C. This follows directly from Gaussian assumptions faces. Of the faces, 150 were from famous people and 150 were
about random effects in the model. The second two terms represent from unfamiliar (previously unseen) people. Scrambled faces were
the departure or divergence of the hyperparameters (encoding the created by a 2D Fourier transform of 150 of the face images, from
covariance) from our prior beliefs. This reports the complexity of which the phases were permuted before transforming back to the
the model (i.e., the effective number of parameters that are used image space, and masking by the outline of the original face image.
to explain the data).
Scrambled faces were therefore approximately matched for spatial
Having estimated the hyperparameters, the posterior estimate frequency power density and for size. Each face or scrambled face
(conditional mean) of the sources is given by:
was either repeated immediately or after a lag of 5–15 intervening
items. Trials with famous faces that were not recognized during a
Jˆ = max p(J | Y) = PY
P = C(2)LT C −1 (6) subsequent debriefing (approximately 39 on average) were ignored
J
as invalid trials, as were trials involving unfamiliar faces incorrectly
where P is the inverse operator, which is evaluated at the conditional labeled as famous during debriefing (approximately 25 on average).
α). For more precise description There were thus nine types of valid trial in total (initial presentation,
means of the hyperparameters ( α̂
of the algorithm and update rules (in the general case considered immediate repetition, and delayed repetition of each of famous faces,
unfamiliar faces and scrambled faces), but for present purposes, no
in Applications 1–3 below), see Figure 2.
Note that the free-energy (Eq. 5) is only a function of the mean distinction was made between initial and repeated presentations, or
and covariance of the hyperparameters (empirical prior source between famous and unfamiliar faces, resulting in just two condicovariances) and not the parameters (source solution), which are tions of interest: all valid face trials and all scrambled face trials. Each
essentially eliminated from the free-energy cost function. This is subject performed the experiment twice (several days apart); once
crucial because it means the real inverse problem lies in optimiz- for concurrent MEG + EEG and once for fMRI + MRI.
On the MEG + EEG visit, 900 trials were presented in a pseuing the empirical priors on the sources, not the sources per se; the
source solution is a rather trivial problem that is solved with Eq. dorandom order (constrained only by the two types of repetition
6, once the hard problem has been solved. Classical inverse solu- lag), split equally across six sessions of ∼7.5 min. The mean reactions (e.g., MMN and beamforming) use Eq. 6 and ignore the hard tion times were 894 ms for faces and 912 ms for scrambled faces.
problem by making strong assumptions about the prior covariance For the fMRI + MRI session, the 900 trials were divided into nine
(i.e., by assuming full priors). By using a hierarchical model, these sessions of 7 min, and within each session, blocks of ∼18 trials were
priors become empirical and can be optimized under parametric separated by 20 s of passive fixation (in order to estimate baseline
(Gaussian) assumptions. This allows us to extend the optimization levels of activity).
of spatial priors to include multiple modalities and subjects; something that classical solutions cannot address (optimally). Before
extending this PEB framework to multi-modal and multi-subject MEG + EEG Acquisition
integration, we introduce the example dataset used to illustrate The MEG data were recorded with a VectorView system (Elekta
Neuromag, Helsinki, Finland), with a magnetometer and two
this integration.
orthogonal planar gradiometers located at 102 positions within
a hemispherical array in a light Elekta-Neuromag magnetically
Methods and procedure
shielded room. The position of the head relative to the sensor array
Subjects and experimental design
was monitored continuously by feeding sinusoidal currents (293–
Eighteen healthy young adults (eight female) were drawn from the 321 Hz) into four head-position indicator (HPI) coils attached
MRC Cognition and Brain Sciences unit Volunteer Panel. The study to the scalp. The simultaneous EEG was recorded from 70 Ag–
protocol was approved by the local ethics review board (CPREC AgCl electrodes in an elastic cap (EASYCAP GmbH, Herrschingreference 2005.08). The paradigm was similar to that used previ- Breitbrunn, Germany) according to the extended 10–10% system
ously under EEG, MEG, and fMRI (Henson, et al., 2003; Henson and using a nose electrode as the recording reference. Vertical and
et al., 2009a). A central fixation cross (presented for a random horizontal EOG (and ECG) were also recorded. All data were samduration of 400–600 ms) was followed by a face or scrambled face pled at 1.1 kHz with a low-pass filter of 350 Hz. Subjects were
(presented for a random duration of 800–1000 ms), followed by seated, and viewed the stimuli that were projected onto a screen
a central circle for 1700 ms. The faces/scrambled faces subtended ∼1.33 m from them.
( )
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Henson et al.
Figure 2 | Full pseudocode description for multi-subject, multi-modal MNM
inversion based on spm_eeg_invert.m in SPM8. For details of the ReML
algorithm and its precise updates using a Fisher-Scoring ascent on free-energy cost
function, see Friston et al. (2008). (Note that an alternative “greedy search”
algorithm is used for maximizing the free-energy in the case of MSP inversions.)
The two main stages are: (1) optimizing the source component hyperparameters
over subjects (by re-referencing each subject’s gain matrix and data to an “average”
space), and (2) optimizing the sensor and source hyperparameters separately for
each subject, using the single optimized source component from stage 1. For a
single subject, the first stage has negligible effect. Note also that this example
assumes a single set of k trials: the full code allows for different numbers of trials
for different numbers of conditions. We have also ignored any filtering or temporal
whitening of the data (see Friston et al., 2006, for more details). + refers to the
pseudoinverse; catj refers to the vertical concatenation of a vector or matrix along
the j-th dimension; tr refers to the trace of a matrix; X = svd(Y, m) refers to a
single-value decomposition of the matrix Y in order to produce the matrix X
containing the m singular vectors with the highest singular values. Note that these
update rules are generic – i.e., apply to all Applications in this paper – all that
changed across Applications was the choice of data (Y) and covariance components
(Q), as illustrated in the generative models shown in subsequent Figures.
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Henson et al.
A 3D digitizer (Fastrak Polhemus Inc., Colchester, VA, USA) was
used to record the locations of the EEG electrodes, the HPI coils
and approximately 50–100 “head points” along the scalp, relative
to three anatomical fiducials (the nasion and left and right preauricular points).
fMRI + MRI Acquisition
MRI data were acquired on a 3T Trio (Siemens, Erlangen, Germany).
Subjects viewed the stimuli via a mirror and back-projected screen.
A T1-weighted structural volume was acquired with GRAPPA 3D
MPRAGE sequence (TR = 2250 ms; TE = 2.99 ms; flip-angle = 9°;
acceleration factor = 2) with 1 mm isotropic voxels. Two FLASH
sequences, and a DWI sequence, were also acquired, but are not utilized here. The fMRI volumes comprised 33 T2-weighted transverse
echoplanar images (EPI; 64 mm × 64 mm, 3 mm × 3 mm pixels,
TE = 30 ms) per volume, with blood oxygenation level dependent
(BOLD) contrast. EPIs comprised 3 mm thick axial slices taken
every 3.75 mm, acquired sequentially in a descending direction. A
total of 210 volumes were collected continuously with a repetition
time (TR) of 2000 ms. The first five volumes discarded to allow for
equilibration effects.
These data will be available for download from http://central.
xnat.org as part of the BioMag2010 data competition (Wakeman
and Henson, 2010).
MEG + EEG pre-processing
External noise was removed from the MEG data using the temporal extension of signal-space separation (SSS; Taulu et al., 2005)
as implemented with the MaxFilter software Version 2.0 (ElektaNeuromag), using a moving window of 4 s and correlation coefficient of 0.98 (resulting in removal of 0–9 components, with an
average of six across participants/windows). The MEG data were
also compensated for movement every 200 ms within each session.
The total translation between first and last sessions ranged from
0.74–9.39 mm across subjects (median = 3.34 mm).
Manual inspection identified a small number of bad channels:
numbers ranged across subjects from 0–12 (median of 1) in the
case of MEG, and 0–7 (median of 1) in the case of EEG. These
were recreated by MaxFilter in the case of MEG, but rejected in
the case of EEG. After reading data from all three sensor-types
into a single SPM8 data format file3, the data were epoched from
−500 to +1000 ms relative to stimulus onset (adjusting for the
34-ms projector delay), adjusted for the mean across the −500
to 0-ms baseline, and down-sampled to 250 Hz (using an antialiasing low-pass filter of approximately 100 Hz). Epochs in which
the EOG exceeded 150 μV were rejected (number of rejected
epochs ranged from 0 to 311 across subjects, median = 76), leaving approximately 400 valid face trials and 245 valid scrambled
trials on average. The EEG data were re-referenced to the average
over non-bad channels.
It is these epoched, pre-processed datasets that were used in
the three PEB applications below. For simplicity, only the magnetometer data are reported here; i.e., “MEG” refers to the 102
magnetometer channels (results from planar gradiometers can be
obtained from the authors; see also Henson et al., 2009a).
3
http://www.fil.ion.ucl.ac.uk/spm
In order to determine a time–frequency window on which to
focus for the source analysis (i.e., to select data features of interest), a time–frequency analysis was performed over the sensors
using fifth-order Morlet wavelets from 6 to 90 Hz, centered every
2 Hz and 20 ms. The logarithm of the resulting power estimates
was baseline-corrected (by subtracting the mean log-transformed
power at each frequency during the −500 to 0-ms period). The 2D
time–frequency data were then averaged over trials and channels
(of a given modality) and analyzed with statistical parametric mapping (SPM). We used a paired t-test of the mean difference between
faces and scrambled faces across subjects, with the resulting statistical map corrected for multiple comparisons across peristimulus
times and frequencies using Random Field Theory (Kilner and
Friston, 2010).
Using an initial peak threshold of p < 0.001 (uncorrected), only
one cluster showed greater source energy for faces than scrambled faces that survived correction for its extent within the time–
frequency space in the EEG data, which ranged between 8 and
18 Hz and from 100 to 220 ms (Figure 3A). A similar cluster was
found in the MEG data, but did not quite survive correction for
extent. No reliable increases in power were found for scrambled
faces relative to faces. This face-related power increase coincided
with a general increase in power for faces and scrambled faces relative to pre-stimulus baseline, which was distributed maximally over
posterior EEG sensors and lateral MEG sensors (Figure 3B), and
coincided with an evoked component that peaked around 160 ms
(corresponding to the N170 and M170 for EEG and MEG components respectively, Henson et al., 2010).
MRI + fMRI pre-processing
Analysis of all MRI data was performed with SPM8. The T1-weighted
MRI data were segmented and spatially normalized to gray matter, white matter, and CSF segments of an MNI template brain in
Talairach space (Ashburner and Friston, 2005).
After coregistering the T2*-weighted fMRI volumes in space,
and aligning their slices in time, the volumes were normalized using
the parameters from the above T1 normalization, resampled to
3 mm × 3 mm × 3 mm voxels, and smoothed with an isotropic
8 mm full-width-at-half-maximum (FWHM) Gaussian kernel.
The time-series in each voxel were scaled to a grand mean of 100,
averaged over all voxels and volumes.
Statistical analysis was performed using the usual summary
statistic approach to mixed effects modeling. In the first stage,
neural activity was modeled by a delta function at stimulus onset
and the ensuing BOLD response was modeled by convolving these
with a canonical hemodynamic response function (HRF) to form
regressors in a general linear model (GLM). Voxel-wise parameter
estimates for the resulting regressors were obtained by maximumlikelihood estimation, treating low-frequency drifts (cut-off 128 s)
as confounds, and modeling any remaining (short-term) temporal
autocorrelation as an autoregressive AR(1) process.
Images of the parameter estimates for each voxel comprised
the summary statistics for the second-stage analysis, which treated
subjects as a random effect. Contrast images of the canonical HRF
parameter estimates for each subject were entered into a repeatedmeasures ANOVA, with the nine conditions (correcting for nonsphericity of the error, Friston et al., 2002). An SPM of the F-statistic
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Henson et al.
Figure 3 | The EEG and MEG data used to illustrate applications of PEB. (A)
Shows a thresholded, 2D Statistical Parametric Map (SPM) for a mass univariate
paired t-test across the 18 subjects (corresponding to the GLM inset) testing for
increases in the (log of the) total energy (induced and evoked) for faces relative to
scrambled faces every 2 Hz between 6 and 90 Hz (y-axis of SPM) and every
20 ms between −500 and +1000 ms (x-axis is SPM) based on a Morlet Wavelet
decomposition of data from each sensor, followed by averaging over these
sensors. The peak threshold corresponds to p < 0.001 uncorrected, with the
extent of the time–frequency region outlined in red for EEG surviving p < 0.05
was then computed to compare the average of the (six) face conditions against that of the (three) scrambled conditions. This SPM{F}
was thresholded for regions of at least 10 contiguous voxels that
survived the peak threshold of p < 0.05 (family-wise error-corrected
across the whole-brain). This analysis produced three clusters, in
right mid-fusiform, right lateral occipital cortex, and a homologous
cluster on the left that encompassed both fusiform and occipital
regions. All these regions evidenced a higher BOLD signal for faces
than scrambled faces (reported later in Figure 6B). These clusters are
in general agreement with previous fMRI studies reporting a similar
contrast of faces versus non-face objects (e.g., a subset of Henson
et al., 2003), most likely corresponding to what have been called the
“FFA” and “OFA” (Kanwisher et al., 1997). After a nearest-neighbor
projection to the vertices in the template cortical mesh (which has a
one-to-one correspondence with vertices in each s ubject’s canonical
(corrected) for multiple comparisons, using Random Field Theory (a similar region
can be seen in the MEG data). (B) Shows time–frequency plots of total
log-energy separately for each sensor for the average response to faces and
scrambled faces. Inset is the trial-averaged evoked response for the sensors
[circled in (B)] that showed maximal energy increases versus baseline, with the
blue line corresponding to faces and the green line corresponding to scrambled
faces (i.e., showing the evoked N/M170 component that is likely to be the
dominant component of the energy increase between 8 and 18 Hz, 100 and
220 ms, for faces relative to scrambled faces).
mesh; see below), the F-values in each cluster were binarized to form
three prior spatial (diagonal) covariance components with 18, 38,
and 33 non-zero entries (see Results below).
EEG/MEG forward modeling
The inverse of the spatial normalization used to map each subject’s
T1-weighted MRI image to the MNI template was used to warp a
cortical mesh from that template brain back to each subject’s MRI
space (see Mattout et al., 2007 for further details). The resulting
“canonical” mesh was a continuous triangular tessellation of the
gray/white matter interface of the neocortex (excluding cerebellum)
created from a canonical T1-weighted MPRAGE image in MNI
space using FreeSurfer (Fischl et al., 2001). The surface was inflated
to a sphere and down-sampled using octahedra to achieve a mesh
of 8196 vertices (4098 per hemisphere) with a mean inter-vertex
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Henson et al.
spacing of ∼5 mm. The normal to the surface at each vertex was calculated from an estimate of the local curvature of the surrounding
triangles (Dale and Sereno, 1993). The same inverse-normalization
procedure was applied to template inner skull, outer skull and scalp
meshes of 2562 vertices.
The MEG and EEG sensor positions were projected onto each
subject’s MRI space by a rigid-body coregistration based on minimizing the sum of squared differences between the digitized fiducials and the manually defined fiducials on the subject’s MRI, and
between the digitized head points and the canonical scalp mesh
(excluding head points below the nasion, given absence of the nose
on the T1-weighted MRI). The gain matrix (L) was then created
within SPM8 by calls to FieldTrip functions4, using a single shell
based on the inner skull mesh for the MEG data (Nolte, 2003), and
a three-shell boundary element model (BEM) based on inner skull,
outer skull and scalp meshes for the EEG data (Fuchs et al., 2002).
Lead fields for each sensor were calculated for a dipole at each
point in the canonical cortical mesh, oriented normal to that mesh.
EEG/MEG inversion parameters
A spatial dimension reduction was achieved by singular-value
decomposition (SVD) of the outer product of the gain matrix,
with a cut-off of exp(−16) for the normalized eigenvalues (which
retained over 99.9% of the variance). This produced m = 55 − 68
spatial modes across subjects for the MEG (magnetometer) data,
and m = 61 − 69 for the EEG data (see Figure 2).
Based on the suprathreshold SPM results for comparison of
faces versus scrambled faces in the sensor time–frequency analysis
(see above), the data were projected onto a (Hanned) time window
between +100 and +220 ms, and a frequency band from 8 to 18 Hz.
The resulting covariance across sensors was averaged across all trials of both trial-types (faces and scrambled), effectively capturing
induced and evoked power (Friston et al., 2006). A temporal dimension reduction was achieved by an SVD of this mean covariance,
once projected onto the spatial modes, using a cut-off of exp(−8)
for the normalized eigenvalues (which retained over 95% of the
4
http://fieldtrip.fcdonders.nl/
variance; see Figure 2). This resulted in r = 3 temporal modes for
every subject (Friston et al., 2006). Thus the covariance matrices
for each modality of approximately 63 × 63 (reduced sensors × sensors) on average were estimated from approximately 645 × 3 (trials × reduced time points) samples.
For simplicity (and to highlight to benefits of fMRI priors),
we used simple generative models with a single source component (the identity matrix, Q1(2) = I p , i.e., MMN model) for source
reconstruction (Table 1). Having estimated the inverse operator,
P (Eq. 6), the power of the source estimate for each condition
(within the same time–frequency window, averaged across trials) was calculated for each vertex, and smoothed across vertices
using eight iterations of a graph Laplacian. The logarithm of these
smoothed power values was then truncated (to remove very small
values), scaled by the resulting mean value over vertices and trials, and interpolated into a 3D space of 2 mm × 2 mm × 2 mm
voxels. Finally, the 3D images were smoothed with an isotropic
3D Gaussian with a FWHM of 8 mm (to allow for residual intersubject differences). Note that the inverse operator is estimated
from the covariance across sensors over a range of times/frequencies within the time–frequency window of interest (more
precisely, by the temporal modes resulting after SVD of the data
filtered to this window, as above), which can be used to reconstruct
a timecourse for each source; it is only when subsequently performing statistics on power estimates that such source estimates
are averaged across time and frequency to produce a single scalar
value for each source.
Results
In this section we present the results of model inversion using the
exemplar data of the previous section. We focus on three generalizations of the basic model above that enable (symmetric and asymmetric) fusion of different imaging modalities and, finally, the fusion
of data from different subjects. Note that the same optimization
scheme was used for all three Applications (as shown in Figure 2);
the only difference across Applications was the specific choice of
covariance components at source and/or sensor levels, Q(hl ) (as shown
in Table 1).
Table 1 | Summary of the differences in generative model across Applications 1–3.
Application
Q1(2)
Q(2)
2
Q(2)
3
Q(2)
4
(1)
Q11
Q(1)
21
1(2) ,Ω1(2)
1(1) ,Ω1(1)
(1)
(1)
2 ,Ω 2
1
EEG
Ip
–
–
–
In
–
−4, 16
−4, 16
–
MEG
Ip
–
–
–
–
In
−4, 16
–
−4, 16
EEGu
Ip
–
–
–
In
In
−4, 16
−4, 16
+4, 1/16
MEG
Ip
–
–
–
In
In
−4, 16
+4, 1/16
−4, 16
E + MEG
Ip
–
–
–
In
In
−4, 16
−4, 16
−4, 16
2
E + MEG + fMRI
Ip
G1
G2
G3
In
In
−4, 16
−4, 16
−4, 16
3
Group-optimized
Ip
G1
G2
G3
In
In
−4, 16
−4, 16
−4, 16
E + MEG + fMRI

Q
u
Ip, In represent p × p, n × n identity matrices, given p sources and n sensors. Gh represents a p × p matrix whose elements are zero except for those on the leading
diagonal that correspond to vertices within the hth fMRI cluster, which are set to one (see Application 2). Q represents a p × p matrix that is a linear combination of
the source covariance components Qh(2) , the linear weightings of which are the hyperparameter estimates from group-optimization (see Application 3). For remaining
symbols, see main text.
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Henson et al.
Application 1: Fusion of EEG and MEG data
The first extension of the basic generative model (Figure 1) is to a
symmetric fusion model of MEG and EEG data (Henson et al., 2009a).
This is shown schematically in Figure 4. Effectively, this corresponds
to concatenating the data, gain matrices, and sensor error terms for
each sensor-type, such that Eq. 1 becomes, for j = 1…d sensor-types:
 Y1   L 1   E1(1) 
     (1) 
 Y2  =  L 2  J +  E 2  (7)
   
     (1) 
 Yd   L d  Ed 
To accommodate different scaling and measurement units across
the different sensor-types, the data and gain matrices are re-scaled
(after projection to the mj spatial modes) as follows:
Y j =
L j =
Yj
1
mj
tr (Yj YjT )
Lj
1
mj
(8)
tr (L j LTj )
where tr(X) is the trace of X. This effectively normalizes the data
so that the average second-order moment (i.e., sample variance if
the data are mean-corrected) of each spatial mode is the average
variance expected under independent and identical sources with
unit variance (ignoring sensor noise). If the gain matrices for each
sensor-type were perfect (and expressed in the same physical units),
this scaling would be redundant. In the absence of such knowledge
however, it allows for arbitrary scaling of the lead fields from different modalities and enables the relative levels of sensor noise to
be estimated for each sensor-type; those levels being proportional
to exp(λλ(j1) ). This is arguably more principled than estimating relative noise levels from, for example, the pre-stimulus period (e.g.,
Molins et al., 2007), which does not discount brain “noise” from
the sources5. For tests of the validity of this scaling, and further
discussion, see Henson et al. (2009a).
Note that the sources in J and their priors (left-hand branch
of Figure 4) are unchanged. It is the sensor-level covariance components (right-hand branches of Figure 4) that are augmented.
Specifically, the sensor error covariance, C(1), becomes:
C1(1) 0 0 



0 C(21)
C(1) = 
 0 


0 C(d1) 
 0
For simplicity, we restrict the present example to just h = 1
sensor–noise component per sensor-type, equal to the identity
matrix (i.e., isotropic or white-noise), Q(j1) = In j ⇒ C(j1) = exp(λ (j1) )Ιn j .
Because the model evidence is conditional on the data, one
cannot evaluate the advantage of fusing MEG and EEG simply
by comparing the model evidence for the fused model (Figure 4)
relative to that for a model of the MEG or the EEG data alone
(Figure 1). Rather, one can fit the concatenated (MEG and EEG)
data, and compare the model evidence for “unimodal” versus
“bimodal” models by varying the sensor-level hyperpriors for each
sensor-type (Eq. 4). If we increase the hyperprior mean for the j-th
sensor-type to (j1) = +4 (i.e., a proportional increase in the prior
noise variance of exp(8)≈3000) and decrease its hyperprior variance
to ω(j1) = 1/16 (i.e., tell the model we are confident the j-th modality is largely noise), then we are effectively discounting data from
that sensor-type. This is because the noise for that sensor-type is
modeled as very high relative to the other modality. Because such
hyperpriors are part of the model, their veracity can be compared
using the free-energy approximation to the log-evidence in the
usual way. In other words, if the j-th modality is uninformative,
then the “unimodal” model for the other modality will have more
evidence than the “bimodal model.” More specifically, we can use
the free-energy (Eq. 5) to test whether a “bimodal” fusion model
with symmetric and weak shrinkage hyperpriors (i.e., η(j1) = −4 and
ω(j1) = 16 for j = 1 and j = 2) has more evidence than either of the
Symmetric multimodal (E+MEG) fusion model
,
Q(2)
h
((1))
1h
((2))
h
Fixed
...
C1((1))
J
Q(1)
2h
((1))
2h
Variable
Y2
Y1
)
C((1)
2
2
1
Y1
...
EEG data
L2
Y2
MEG data
C (j1) = ∑ h  (jh1)Q(jh1)
For MEG data, an estimate of sensor–noise covariance can be obtained from empty-room recordings (Henson et al., 2009a), for which the noise consists of a random
component intrinsic to the SQUID sensors and associated electronics, plus a component from environmental magnetic noise (that has not been fully attenuated by
magnetic shielding). Methods for estimating such sensor–noise covariance for EEG
data are less obvious however.
Q1(1)h
C((2))
L1
where C(j1) for each sensor-type j is again estimated by a linear
combination of covariance components:
5
...
Figure 4 | The extension of the generative model in Figure 1 to the
fusion of EEG and MEG data, i.e., j = 1–2 modalities (Y1 and Y2). This model
was the one used to produce the results in Figure 5; i.e., with one minimum
norm source prior (Q1(2) = Ip , where Ip is a p × p identity matrix for the p
sources) and one white-noise sensor component for each modality (Q(j1) = Inj ,
where nj is the number of sensors for the j-th sensor-type).
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Henson et al.
two “unimodal” models that effectively discount the other modality
with an asymmetric noise hyperprior ( (j1) = +4 and ω(j1) = 1 /16 for
j = 1 or j = 2). See Table 1 for a summary of the different models
used in this Application.
The resulting free-energies for each of these three models
– unimodal EEG (MEG discounted), unimodal MEG (EEG discounted) or bimodal EEG + MEG are shown in the leftmost
panel of Figure 5. In nearly all subjects (shown by lines), the
bimodal (fused) model has a higher evidence than either unimodal model, as confirmed by two-tailed, paired t-tests relative to unimodal EEG: t(17) = 3.70, p < 0.005, and relative to
unimodal MEG, t(17) = 2.98, p < 0.01. This advantage of fusion
was manifest both in improved model accuracy (middle panel),
one-tailed t > 2.71, p < 0.05, and reduced model complexity
(rightmost panel), one-tailed t > 1.88, p < 0.05, relative to both
unimodal models. These results complement and extend our
previous evaluation of MEG + EEG fusion, where we examined
the change in the posterior precision of the source estimates
(Henson et al., 2009a).
The reconstructed sources also show differences between the
separate and fused inversions (Figure 6A). The mean source power
across subjects on the MNI cortical surface is similar for all three
models (see maximum intensity projections – MIPs – inset in the
top right of each column), with a predominance of power in bilateral occipito-temporal cortex, especially on the right. The reliability
of this pattern across subjects, as reflected by t-values surviving
p < 0.001 uncorrected within the MNI volumetric space (the main
MIPs in each column), were more different across inversions however, being greater on the left in the case of EEG alone and greater
on the right in the case of MEG alone. Importantly however, the
t-values for the bimodal model recovered a bilateral pattern, which
also spread more anteriorly along the ventral surface of the temporal lobe (consistent with the fMRI data in Figure 6B). Indeed, the
maxima in left (−30 −60 −14) and right (+42 −72 −14) fusiform
survived correction for multiple comparisons across the volume,
t(17) > 6.40, p < 0.05 corrected. This more plausible source reconstruction following fusion of EEG and MEG echoes that found
when using MSP in Henson et al. (2009a).
Application 2: Asymmetrical Integration of EEG and MEG data
with fMRI data
The second extension of the basic generative model is to an integration of MEG and EEG with fMRI (Henson et al., 2010), as shown in
Figure 7. This corresponds to asymmetric multi-modal integration
(as distinct from the symmetric integration in the previous section;
Daunizeau et al., 2005), in the sense that the fMRI data are not fit
simultaneously with the MEG and EEG data (i.e., do not appear
at the bottom of the dependency graph in Figure 7), but rather are
used to define the prior covariance components on the sources (i.e.,
the fMRI data correspond to the third data type, Y0, at the top of
the graph). The reason for this is reviewed later.
To map fMRI data to a (small) number of covariance compo2)
in Figure 7), the typical procedure is to threshold an
nents (Q(h>1
SPM of the relevant contrast of fMRI data to produce a number
of clusters (contiguous suprathreshold voxels). These clusters
can then be projected to the nearest corresponding vertices in
each subject’s cortical mesh. Here the canonical cortical mesh
(Mattout et al., 2007) is useful, because this provides a one-to‑one
mapping between the vertices of each subject’s cortical mesh
and the vertices in standard (e.g., MNI) space. This allows the
fMRI clusters to be defined by different subjects, or, as here, by
group statistics on spatially normalized fMRI data across subjects. Figure 6B shows a MIP of clusters of at least 10 voxels,
whose peak statistical values survived p < 0.05 (corrected) for
multiple comparisons across the brain, where that statistic is the
F-value for a two-tailed6 paired t-test of faces versus scrambled
faces across the same 18 subjects whose EEG/MEG data are considered here. This furnished three clusters, in right fusiform and
right lateral occipital cortex, plus a single cluster encompassing
similar regions on the left. Each cluster gives a vector, qh, across
vertices for the h-th cluster, whose values are the interpolated
fMRI activation values for each vertex within that cluster, and
zero otherwise. The corresponding source prior covariance component is then:
Q(h2) = G ( f (q h ) f (q h ))
T
where G(·) is some matrix function, and f(x) is some scalar “linkage”
function (linking fMRI activation values, which could be statistic
values or % signal change, to EEG/MEG prior covariance terms).
Here, we use a Heaviside linkage function corresponding to f(x) = 1
for x > 0, and f(x) = 0 otherwise, and G(·) = diag(·) (which puts the
values of f on the leading diagonal of a matrix), since such simple
“binary, variance” priors were sufficient for similar data in Henson
et al. (2010). In other words, the prior variance for all vertices within
a cluster was equal (regardless of the F-value of the corresponding
voxels in the fMRI analysis), and the prior covariance between vertices was zero [positive covariance could be introduced by setting
G(·) to the identity function; i.e., defining covariance components
by the outer product of f(qh)].
An important aspect of this approach to integration of fMRI with
EEG/MEG is that each fMRI cluster contributes a distinct covariance component; i.e., each cluster can be up- or down-weighted
by its own hyperparameter,  (h2) (for further discussion and evidence for this claim, see Henson et al., 2010). This allows for the fact
that the neural activity giving rise to the BOLD data might not be
expressed proportionally in E/MEG data. Note that this approach
is fundamentally different from assuming (ad hoc) fixed values for
the variance of source activity in these regions (cf, Liu et al., 1998),
but is functionally similar to having a separate hyperprior for the
variance (or precision) at each source location, and increasing the
mean (or dispersion) of that hyperprior for locations corresponding
6
A two-tailed test was used in order to make minimal assumptions about the
mapping between BOLD and EEG/MEG signals (Henson et al., 2010; though in
the present data, all three clusters did show a greater BOLD signal for faces than
scrambled faces). Note also that we are using fMRI priors that are defined by the
difference between faces versus scrambled faces, yet inverting the E/MEG data after
collapsing across both faces and scrambled faces (and only contrasting faces versus
scrambled faces after estimating the sources). This is a subtle but important point
that we discuss in detail elsewhere (Henson et al., 2007): In brief, this rests on the
assumption that the generators of the power (within our time–frequency window)
induced by faces and scrambled faces versus pre-stimulus baseline coincide with
those that show differences in such power for faces versus scrambled faces. One
measure of the extent to which this is a valid assumption is the free-energy: if this
assumption were false, we would not anticipate an increase in free-energy when
adding the fMRI priors (Henson et al., 2010).
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Henson et al.
Figure 5 | The bars in the leftmost plot show the mean across subjects of
the maximized free-energy (F) bound on the model log-evidence (arbitrary
units) for (1) “unimodal” EEG model (EEGu), (2) “unimodal” MEG model
(MEGu), and (3) bimodal “fusion” model (E + MEG). All models explain the
same concatenated EEG and MEG data (projected to the time–frequency window
identified in Figure 3), but “unimodal” models have a sensor-level hyperprior with
to s ignificant fMRI signal (Sato et al., 2004)7. In other words, each
fMRI activation locus may, after the hyperparameters have been
optimized, show a greater or smaller variance than other components not based upon fMRI. One important example of this dissociation between fMRI and electromagnetic activation would be when
the fMRI data in one or more regions reflects neural activity arising
before or after the time window of E/MEG data modeled (given the
much slower dynamics of the BOLD signal). For example, it is possible that some of the clusters in the present fMRI data reflect neural
activity that is related to face processing but outside our 100–220 ms
time window (see Henson et al., 2010, for further discussion). The
ability of this approach to discount such “invalid” spatial priors has
previously been demonstrated by simulation (Phillips et al., 2005;
Mattout et al., 2006).
For real data, the effect of adding the three fMRI source priors
to the basic MMN prior (the identity matrix Q1(2) = I p) on the freeenergy for each combination of sensor-type – EEG data alone, MEG
data alone, or fused EEG + MEG data – is shown in Figure 8A.
In nearly all subjects, the addition of the fMRI priors increased
the free-energy significantly for each sensor-type, resulting in a
significant improvement on average, t(17) > 3.46, p < 0.005. The
impact on source reconstruction for the fused EEG + MEG case
is shown in Figure 6C, where it can be seen that the fMRI priors
have “pulled” the source energy to the right fusiform region, in
terms of both the mean source energy on the surface (inset) and
group-level statistics in volumetric space. Indeed, while there are
7
There are important differences between the present approach and that of Sato
et al. (2004): we optimize covariance components (not precisions), and use fMRI
to add formal priors (in terms of components) as opposed to modulating the mean
of the hyperpriors. Our approach does not suppress signal from non-fMRI areas.
It would be an interesting future project to compare the two approaches formally.
For present purposes, the similarity of the two approaches is more relevant, in that
both provide a principled source-specific soft constraint on the reconstruction, based on fMRI.
a high mean and precision for the other modality, which effectively discounts data
of that modality as noise (see text). The circles joined by lines show the results for
each individual subject. The middle and rightmost plots show the two parts of the
free-energy: the accuracy (Fa) and the model complexity (Fc); see text for details
(note that these differ by constant terms that not included here). The advantage of
EEG and MEG fusion is reflected by the increase in F and Fa and decrease in Fc.
no longer any maxima that survive whole-brain correction, the
t-value of the right fusiform maximum with fMRI priors (+40
−42 −22) increased from 4.21 (without fMRI priors) to 4.68 (with
fMRI priors). Note also that the mean source energy is not only
more focal, relative to the model without fMRI priors (inset in
rightmost panel of Figure 6A), but also more anterior, i.e., “deeper”
(further from the sensors). This shows how fMRI priors can overcome the well-known bias of the MMN solution toward diffuse
superficial source estimates (the reason may be due in part to the
sparsity afforded by discrete, compact priors, in the same way that
others have shown how sparse priors reduce the superficial bias
of L2-norm based inverse solutions; Trujillo-Barreto et al., 2004;
Friston et al., 2008). Note however that the “sparseness” of such
fMRI priors will depend on the threshold used to define the fMRI
clusters (e.g., we would obtain fewer and larger clusters had we
reduced the statistical threshold)8.
A symmetrical fMRI + E/MEG fusion model?
It is worth noting at this point that one could also consider a symmetric version of E/MEG and fMRI integration, one possible example of which shown in Figure 9 (see also Luessi et al., 2011). Here, a
common set of electrical sources (J) generate both the E/MEG and
8
It is interesting to consider how the effect of fMRI priors might depend on their relative “size” (spatial extent – i.e., number of non-zero elements). In general, their effect is likely to depend on the data. If the true sources overlap with only one of those
components, then its hyperparameter is still likely to be greater than those of other
components, even if those components have a larger spatial extent, unless there is a
high correlation in the mean gain vectors associated with each cluster of vertices. In
the latter case, it is possible that “bigger” components will be favored, because their
hyperparameter estimate can remain smaller (less far from its shrinkage hyperprior
mean) while still fitting the data as well. The same is likely to apply to covariance
components with comparable spatial extent, but higher magnitudes of non-zero
elements (i.e., higher prior variances on the source parameters), as might happen
for example if the covariance components were a continuous (rather than binary,
as here) function of the underlying fMRI statistic or fMRI signal.
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Henson et al.
Figure 6 | Maximal Intensity Projections (MIPs) in MNI space for increases
in the log-energy for faces, relative to scrambled faces, within a 100 to
220-ms, 8–18 Hz time–frequency window. The MIPs inset to the top right of
each panel show the average increase in this face-related energy across
subjects for the 512 vertices showing the maximal such increase. The main
MIPs in each panel represent SPMs of the corresponding t-statistic (thresholded
at p < 0.001 uncorrected), after interpolating the cortical source energies for each
subject into a 3D volume. (A) shows (from left to right) the results for inverting
fMRI data. In this case, the (spatial) forward model for the fMRI data
(L0) is relatively simple; e.g., a spatial smoothing kernel that depends
on the spatial dispersion of the BOLD signal and spatial resolution
EEG data alone, MEG data alone, or from fusing EEG and MEG data in the
“bimodal” model of Figures 4–5. (B) shows the results of the same contrast
but on the fMRI data (though two-tailed, and at a higher threshold of p < 0.05
corrected for multiple comparisons). (C) shows the results of inverting EEG and
MEG data after the addition of three fMRI priors [those in (B)] using the
generative model shown in Figure 7. (D) Shows the results of groupoptimization of the fMRI priors used in (C), using the generative model shown in
Figure 10.
(voxel-size) of the fMRI data (relative to the density of the electrical sources modeled). The separate error component for the fMRI
data can also be relatively simple, e.g., a similar spatial smoothness
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Henson et al.
Asymmetric (E+MEG+fMRI) multimodal integration model
Y0
Q1(1)h
((1))
1h
((2))
h
C((2))
J
L1
,
,
...
Q1(2) Q(2)
2
Possible symmetric (M/EEG+fMRI) multimodal fusion model
...
C1((1))
((1))
2h
)
C((1)
2
...
Q(0)
h
Y2
Variable
Y1/2
M/EEG data
L0
L2
Y0
((0))
h
...
Q(1)
h
(2)
h
C(2)
((1))
h
J
0
Fixed
Fixed
Q(2)
h
C((0))
2
1
Y1
...
Q(1)
2h
C((1))
1
Y1
Y0
Variable
...
Y1
M/EEG data
L1
Y0
fMRI data
fMRI data
Source and sensor/voxel space
Figure 7 | The extension of the generative model in Figure 4 to the
addition of fMRI priors (Y0).
Figure 8 | The bars show the difference in the mean across subjects of
the maximized free-energy bound on the model log-evidence (arbitrary
units) for two models: with versus without the three fMRI priors (A), or
with versus without group-optimization of those priors (B), for EEG data
alone, MEG data alone, or the combined EEG and MEG data. The circles
show values for each subject’s difference. fMRI priors significantly improve
the mean free-energy, but their group-optimization does not (as expected, see
text). F(1) refers to the maximal free-energy from the first stage of groupoptimization of the fMRI source hyperparameters, whereas the F(2) refers to
the maximal free-energy from the second stage of individual inversion of each
subject when using the group-optimized source prior (see Figure 2).
kernel (Flandin and Penny, 2007). Note that the fMRI kernel matrix
L0 would generally have a higher ratio of rows to columns than the
E/MEG gain matrix L1; reflecting the typically greater number of
fMRI measurements (voxels) than E/MEG measurements (for a
single time point). In other words, the indeterminacy in the spatial
mapping from electrical sources to measurements is typically much
Figure 9 | A possible symmetric model for fusion of EEG/MEG data and
fMRI data (cf. Figure 7).
less for fMRI than E/MEG, which means that the fMRI data are likely
to have the dominant effect on the estimation of the location of
electrical sources. This likely dominance of the fMRI data therefore
questions the value of such a symmetric fusion model. Furthermore,
there are potential problems that arise when electrical sources do
not produce detectable fMRI signals and vice versa.
One common approach to the different spatial scales of fMRI
and M/EEG is to make both the fMRI data (Y0) and M/EEG source
estimates (J) depend on a third, latent variable (e.g., Trujillo-Barreto
et al., 2001; Daunizeau et al., 2007; Ou et al., 2010). This variable
can have a spatial prior (e.g., for smoothness, or for sparseness) that
encourages a common spatial covariance structure across fMRI and
M/EEG parameters (with possibly different temporal profiles). This
alleviates the problems associated with having the current estimates
J generate both the fMRI data and the M/EEG sensor data that is
depicted in Figure 9. However, the value of such models is still
unclear when one considers the vastly different temporal resolutions of the two modalities. fMRI data are typically only sampled
every few seconds, and are themselves the product of a temporal
smoothing of electrical activity by the HRF. Therefore a full spatiotemporal multi-modal fusion model would entail a forward model
for the fMRI data that took into account hemodynamic smoothing.
This could even take the form of a biophysical model with many
parameters, based on physiological knowledge of hemodynamics
(e.g., Sotero and Trujillo-Barreto, 2008). In this situation, there
would be (temporal) information about the sources in the E/MEG
data that is not present in the fMRI data. For this additional temporal information in the E/MEG data to aid estimation of the spatial
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Henson et al.
distribution of the sources, or for that matter, for the ability of the
additional spatial information in the fMRI data to aid estimation
of the temporal profile of the sources, there would need to be some
dependency between the spatial and temporal parameters controlling the electrical currents in the brain. Unfortunately, there is no
principled reason to think that there will be strong dependencies
of this sort, because there is no evidence that the potential range
of dynamics of electromagnetic sources varies systematically across
different parts of the brain. If the spatial and temporal parameters
of forward models are indeed (largely) conditionally independent,
then the most powerful approach to EEG/MEG and fMRI integration may be asymmetric: i.e., using EEG/MEG data as temporal
constraints in whole-brain fMRI models, or using fMRI data as
spatial priors on the EEG/MEG inverse problem, as considered here.
Application 3: Group-optimization of Source Priors
The final extension of the basic generative model in Figure 1 is to
the group-optimization of source priors, as shown in Figure 10.
For each modality, there are now there are multiple datasets (Yi for
the i-th subject), each explained by separate source distributions
(Ji) with separate error terms (Ei), but sharing the same priors.
The estimation of the hyperparameters is therefore optimized by
pooling data across subjects, as was originally demonstrated using
several hundred sparse priors (MSP, Litvak and Friston, 2008).
Here, we apply this optimization to the four source priors in the
previous section, comprising the MMN constraint plus the three
Multisubject fusion model
Y0
Q1(2) Q(2)
2
C((2))
J1
L1
Y11
Fixed
,
...
...
Q1(1)h
(1)
1jh
((2))
h
J2
12
11
Y12
L2
Variable
(1)
2jh
C1((1))
L1
Yij
Y21
M/EEG data for
jth sensor-type
from ith subject
...
Q(1)
2h
21
Y22
Y0
)
C((1)
2
22
L2
fMRI data
Figure 10 | The extension of the generative model in Figure 7 to
group-optimization of source priors, for i = 1–2 subjects, each with data
from j = 1–2 modalities.
fMRI priors. Starting with the simple case of one modality, the
basic idea for group models is to concatenate the data from i = 1…s
subjects over samples:
 J1 
 A1Y1 ,…, A s Ys  =  A1L1 ,…, A s L s    + E1(1) ,…, E(s1)  (9)
 
 Js 
This contrasts with the model in Eq. 7 where we concatenated
over channels. In multi-modal integration, we consider each new
modality as an additional set of sensors that are picking up signals
generated by the same sources. For multi-subject integration, the
sources may be different but are deployed under the same spatial
priors; because we examine subjects under the assumption that (a
priori) they have the same functional anatomy. This means each
subject’s data constitute a separate realization and should be treated
as a further sample of the same thing. We therefore concatenate
over samples (analogous to concatenating over multiple trials from
one subject; Friston et al., 2006). However, to do this we must realign the subjects’ data so that the effective lead fields are the same
over subjects. This is achieved by a “re-referencing” matrix, Ai, that
projects each subject’s gain matrix to a common or “average” gain
matrix; i.e., AiLi = 〈AiLi〉 (see also Valdés-Hernández et al., 2009, for
a related approach). This average is computed with the constraint
that its preserves the information (on average) in the sensor data:
Ai L i = L : L = Ai L i
i
{
}
 T | : tr (LL
 T ) = n
s .t . : Ai = max arg | LL
This is a complicated non-linear problem that can be solved
using recursive least squares and an implicit generalized eigenvalue
solution (see Figure 2). Clearly, the average lead field cannot necessarily “see” all the data sampled by the original (subject-specific)
lead fields. Typically about 10% of the data is lost, in the sense it lies
in the null space of the average lead field. Nonetheless, this form of
re-referencing is an improvement on that described in Litvak and
Friston (2008), where all the lead fields were re-referenced to the
first AiLi = L1, under the assumption that this was representative of
the remaining lead fields. The solution above is an improvement in
the sense that one does not have to assume the first subject’s lead
field is “representative,” and the re-referencing does not depend on
which subject is designated as the “first.”
Crucially, although the sources Ji are subject-specific, the empirical source priors are the same. This prior is used in the hope that
pooling data across individuals will provide extra constraints on
the hyperparameters (assuming sources are sampled from a spatial
prior that is common to subjects). If this prior assumption is correct, the ensuing subject-specific reconstructions should improve
the consistency of the source localization across subjects (but not
bias any trial-specific differences at any given location).
In practice, group-optimization entails two stages of hyperparameter optimization (see Figure 2): In the first stage, the source
hyperparameters  (h2) are estimated based using Eq. 9. In the secondstage, a single weighted combination of the source priors (weighted
Qh(2) )
by the hyperparameters estimated in the first stage; Sh exp α̂ (2)
h
is combined with the sensor covariance components, and their hyperparameters estimated separately for each subject’s (unreferenced) data
and gain matrix, as in Eq. 7.
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( )
Henson et al.
We can combine multi-modal models (Eq. 7) and multi-subject
models (Eq. 9), as shown in Figure 10, in order to invert joint
models of the form (for d modalities and s subjects)
(1)
 A11Y11 ,…, A1s Y1s   A11L 11 ,…, A1s L 1s   J1  E11
,…, E1(1s) 
 
  

 [10]
=
  + 


 Ad 1Y d 1 ,…, Ads Y ds   Ad 1L d 1 ,…, Ads L ds   Js  E(d11) ,…, E(ds1) 


 


This is the most general form of the model currently supported
by the Matlab routine spm_eeg_invert.m in SPM8.
The effect of group-inversion of the four empirical source priors
on the free-energy for each combination of sensor-type is shown
in Figure 8B. Group-optimization had no significant effect on the
mean free-energy across subjects. This is expected: constraining
the weighting of source priors across subjects gives less scope for
maximizing the model evidence for any single subject (relative to a
model optimized for that subject’s data alone). Indeed, in a previous
application to hundreds of sparse priors, Litvak and Friston (2008)
found a decrease in the mean free-energy after group-inversion.
Rather, the effects of group-optimization are apparent in the statistical tests of the source estimates across subjects, as is apparent in
Figure 6D. Though the mean source estimates appear little affected
(inset in Figure 6D), there are many more voxels with t-values
that survive thresholding (compared to Figure 6C), including a
suprathreshold cluster in the left, as well as right, fusiform. In fact,
group-optimization tripled the number of suprathreshold voxels
(from 1,104 to 3,162) and increased the maximal t-value (in right
fusiform) from 4.67 to 5.91.
The effect of group-optimization on the hyperparameters themselves is shown in Figure 11. The bars constitute a histogram of the
hyperparameter estimates across subjects, for each of the four source
Figure 11 | The hyperparameter estimates (̂(2) ) for the MMN source
prior (top left) and three fMRI source priors (remaining plots; OFA,
occipital face area; FFA, fusiform face area; l, left, r, right). The solid bars
constitute a histogram of estimates across subjects without groupoptimization; the solid line reflects the single estimate after group-optimization
across subjects; and the dotted line shows their prior expectation ((2) = −4) .
Note the different scale of the x-axis in the top right panel.
priors after inverting each subject’s data separately, using Eq. 7. The
solid vertical line is the single group-optimized estimate, and the
dotted line is the hyperprior mean of  = −4. Group-optimization
not only enforces a consistent ratio of hyperparameter estimates
across subjects, but can also increase those estimates above the average without group-constraints. In the case of the lOFA/FFA prior
(lower left panel), for example, the solid line (group-optimized estimate) is to the right of the central tendency of the histogram of individual estimates. Or in the case of the right OFA prior (upper right
panel), its hyperparameter estimate was effectively shrunk to zero
for some subjects without group-optimization, but remains close
to the hyperprior mean (as αˆ i = −3.86) with group-optimization. It
should be remembered that this group-optimization assumes that
the same set of cortical generators exists in all subjects, i.e., that
subjects differ only in the overall scaling of this set (as reflected by
single hyperparameter ˆ i(2) for each subject i in the final stage in
Figure 2), and in their relative sensor-level noise components (as
reflected by the hyperparameters ˆ i(1) in Figure 2). If one does not
wish to make this assumption, then group-optimization need not
be selected, and each subject inverted separately.
Discussion
We have reviewed a theoretical framework called Parametric
Empirical Bayes (PEB) that we believe offers a natural and powerful way to introduce multiple constraints into the inverse problem
of estimating the cortical generators of EEG/MEG data. Empirical
Bayes is a general approach afforded by hierarchical formulation
of linear generative models, which is explicitly or implicitly used
in many other approaches to the EEG/MEG inverse problem (e.g.,
Baillet and Garnero, 1997; Trujillo-Barreto et al., 2001, 2008; Sato
et al., 2004; Daunizeau et al., 2007; Wipf and Nagarajan, 2009; Ou
et al., 2010; Luessi et al., 2011). We have illustrated three applications of PEB to an example dataset: (1) the symmetric integration
(fusion) of MEG and EEG data, which entailed common source
priors (a single MMN prior) but separate modality-specific sensor
error components, (2) the asymmetric integration of EEG/MEG
data with fMRI data, in which significant clusters in the fMRI data
form additional, separate spatial priors on the EEG/MEG sources,
and (3) the optimization of multiple source priors (in this case
from the fMRI clusters) across subjects, by re-referencing their
data and gain matrices to a common source space. The benefit
of these three applications was apparent in improvements in the
variational free-energy bound on the Bayesian log-evidence of the
generative model, and/or improvements in the number or location
of suprathreshold sources in a statistical comparison of spatially
normalized source images across subjects.
One advantage of an empirical (hierarchical) Bayesian framework is that the degree of regularization of the inverse solution
by prior constraints is optimized by virtue of an implicit hierarchical generative model of the data, rather than being fixed
in advance. This optimization refers to the estimation of the
hyperparameters in order to maximize (a bound on) the model
evidence (for further elaboration, see Friston et al., 2007; Wipf and
Nagarajan, 2009). So, in the case of the spatial priors from fMRI
in Application 2 above, rather than assuming a fixed ratio (e.g.,
10%) for the prior variance on sources with and without fMRI
activations (Liu et al., 1998), the hyperparameter c ontrolling the
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Henson et al.
variance of each fMRI prior is estimated anew for each dataset9.
Furthermore, the ability to estimate multiple hyperparameters
– i.e., the influence of multiple prior constraints – is an important advance on traditional approaches to the EEG/MEG inverse
problem. Application 2 described one advantage, in the context of
multiple fMRI priors, when different fMRI priors can contribute
differently to the EEG/MEG data. In particular, if a subset of the
fMRI priors reflect neural activity that is not represented within
the time/frequency window of EEG/MEG data being localized,
then their hyperparameters should shrink to zero. Another example is the use of MSP (Friston et al., 2008), that furnish more focal
solutions. It is the presence of multiple priors that then offers the
possibility to optimize the relative value of their hyperparameters
across subjects, as shown in Application 3.
A further advantage of the PEB framework is that the maximization of variational free-energy not only optimizes the hyperparameters, but its maximum also provides an upper bound on the
model log-evidence. This offers a natural way to compare different
generative models of EEG/MEG data. It can be used, for example, to
compare different electromagnetic forward models (Henson et al.,
2009b), or to optimize the number of equivalent current dipoles
(Kiebel et al., 2008). Its maximization can also be used to optimize
more specific details of the generative model: For example, to explore
the choice of the “linkage function” that relates fMRI values (e.g.,
t-statistic or % signal change) to the values in the source (co)variance
matrices in Application 2 above (Henson et al., 2010). One important issue for future further consideration however, particularly with
many hyperparameters, is the possible existence of local maxima in
the free-energy cost function (Wipf and Nagarajan, 2009).
There are clearly assumptions in this review that deserve further exploration. Some of these are specific to the above applications: For example (1) the particular type of scaling used for
9
The prior mean and variance of the hyperparameters (the hyperpriors), on the
other hand, are fixed. Nonetheless, they are based on principled reasons, namely
to provide weak shrinkage (see, for example, Application 1), and could in principle
be explored by further maximization of the variational free-energy (see Henson
et al., 2007).
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Available at: http://www.frontiersin.
org/Community/AbstractDetails.
aspx?ABS_DOI=10.3389/conf.
fnins.2010.06.00404
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Conflict of Interest Statement: The
authors declare that the research was conducted in the absence of any commercial
or financial relationships that could be
construed as a potential conflict of interest.
Received: 01 February 2011; paper pending
published: 15 April 2011; accepted: 21 July
2011; published online: 24 August 2011.
Citation: Henson RN, Wakeman DG,
Litvak V and Friston KJ (2011) A parametric empirical Bayesian framework for
the EEG/MEG inverse problem: generative
models for multi-subject and multi-modal
integration. Front. Hum. Neurosci. 5:76.
doi: 10.3389/fnhum.2011.00076
Copyright © 2011 Henson, Wakeman,
Litvak and Friston. This is an open-access
article subject to a non-exclusive license
between the authors and Frontiers Media
SA, which permits use, distribution and
reproduction in other forums, provided the
original authors and source are credited and
other Frontiers conditions are complied with.

Henson et al, 2011 - Wellcome Trust Centre for Neuroimaging

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